International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 210211
Section 1.7.3.3.4. Sumfrequency generation (SFG)^{a}Institut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and ^{b}Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France 
SHG () and SFG () are particular cases of threewave SFG. We consider here the general situation where the two incident beams at ω_{1} and ω_{2}, with , interact with the generated beam at ω_{3}, with , as shown in Fig. 1.7.3.17. The phasematching configurations are given in Table 1.7.3.1.

Frequency upconversion process . The beam at ω_{1} is mixed with the beam at ω_{2} in the nonlinear crystal NLC in order to generate a beam at ω_{3}. are the different powers. 
From the general point of view, SFG is a frequency upconversion parametric process which is used for the conversion of laser beams at low circular frequency: for example, conversion of infrared to visible radiation.
The resolution of system (1.7.3.22) leads to Jacobian elliptic functions if the waves at ω_{1} and ω_{2} are both depleted. The calculation is simplified in two particular situations which are often encountered: on the one hand undepletion for the waves at ω_{1} and ω_{2}, and on the other hand depletion of only one wave at ω_{1} or ω_{2}. For the following, we consider plane waves which propagate in a direction without walkoff so we consider a single wave frame; the energy distribution is assumed to be flat, so the three beams have the same radius w_{o}.
The resolution of system (1.7.3.22) with , , and , followed by integration over , leads towithin the same units as equation (1.7.3.70).
or .
The undepleted wave at ω_{p}, the pump, is mixed in the nonlinear crystal with the depleted wave at ω_{s}, the signal, in order to generate the idler wave at . The integrations of the coupled amplitude equations over () with , , and givewith and , whereThus, even if the upconversion process is phasematched (), the power transfers are periodic: the photon transfer efficiency is then 100% for , where m is an integer, which allows a maximum power gain for the idler. A nonlinear crystal with length is sufficient for an optimized device.
For a small conversion efficiency, i.e. ΓL weak, (1.7.3.85) and (1.7.3.86) becomeand The expression for P_{i}(L) with is then equivalent to (1.7.3.83) with or , and or .
For example, the frequency upconversion interaction can be of great interest for the detection of a signal, ω_{s}, comprising IR radiation with a strong divergence and a wide spectral bandwidth. In this case, the achievement of a good conversion efficiency, P_{i}(L)/P_{s}(0), requires both wide spectral and angular acceptance bandwidths with respect to the signal. The double noncriticality in frequency and angle (DNPM) can then be used with onebeam noncritical noncollinear phase matching (OBNC) associated with vectorial group phase matching (VGPM) (Dolinchuk et al., 1994): this corresponds to the equality of the absolute magnitudes and directions of the signal and idler group velocity vectors i.e. .
References
Dolinchuk, S. G., Kornienko, N. E. & Zadorozhnii, V. I. (1994). Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared upconversion. Infrared Phys. Technol. 35(7), 881–895.